N u c l e a r P h y s i c s B 3 1 1 ( 1 9 8 8 / 8 9 ) 1 7 1 - 1 9 0 N o r t h - H o l l a n d , A m s t e r d a m R E C E N T P R O G R E S S I N T H E T H E O R Y O F N O N C R I T I C A L S T R I N G S V . A . K A Z A K O V

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Nuclear Physics B311 (1988/89) 171-190 North-Holland, Amsterdam



The Niels Bohr Institute, University of Copenhagen, Blegdamsvej 17, DK-2100 Copenhagen, Denmark and

Cybernetics Council, Academy of Science, ul Vavilova 40, 117333 Moscow, USSR*

and A. A. MIGDAL

Physics Dept. B-O19, UCSD LaJolla, CA 92093, USA**


Cybernetics Council, Academy of Science. ul Vavilova 40, 117333 Moscow, USSR

Received 20 May 1988

We compare the results of analytical and numerical studies of lattice 2D quantum gravity, where the internal quantum metric is described by random (dynamical) triangulation, with the recent results of conformal approach developed by Knizhnik, Polyakov and Zamolodchikov. The remarkable agreement is underlined for the interactions of gravity with matter fields: Ports spins, D-dimensional Gaussian fields (bosonic string). Some new results are presented for D = 1 discretized bosonic strings satisfying the predictions of conformal theory for the critical expo- nents: "&tr = 0, ~'str = 0, but with unusual logarithmic corrections.

1. Introduction

I n the l a s t few y e a r s the m a i n interest o f specialists in f u n d a m e n t a l p h y s i c s has c o n c e n t r a t e d o n the t h e o r y of strings a n d s u p e r s t r i n g s in critical d i m e n s i o n a l i t y of s p a c e t i m e . T h i s t h e o r y is c o n s i d e r e d as a g o o d c a n d i d a t e for the u n i f i e d t h e o r y of all i n t e r a c t i o n s o f n a t u r e . Because o f success of critical strings a n d p e r h a p s b e c a u s e of d i f f i c u l t i e s o f i n v e s t i g a t i o n of strings in n o n c r i t i c a l d i m e n s i o n s the l a t t e r were n e a r l y f o r g o t t e n .

H o w e v e r , the i n v e s t i g a t i o n s of n o n c r i t i c a l strings were c o n t i n u e d b y a n u m b e r of p e o p l e . T h e s e i n v e s t i g a t i o n s have n o w two m a i n d i r e c t i o n s : l a t t i c e a p p r o a c h a n d c o n f o r m a l field t h e o r y a p p r o a c h .

T h e first i d e a s o f lattice a p p r o a c h were l a i d o u t in refs. [1 3], where s o m e first v e r s i o n s o f d y n a m i c a l l y t r i a n g u l a t e d r a n d o m surfaces were f o r m u l a t e d , S o m e other i m p l e m e n t a t i o n s of Regge calculus in t w o - d i m e n s i o n a l g r a v i t y c a n b e f o u n d in refs.


* Address after July 3, 1988

** Present address

0550-3213/88/$03.50©Elsevier Science Publishers B.V.

(North-Holland Physics Publishing Division)


172 V.A. Kazakoo, A.A. Migdal / Noncritical strings

T h e final formulation of the lattice model of Polyakov's bosonic string and an understanding of the role of dynamical triangulations as an analog of quantum fluctuations of the internal metric was achieved in refs. [6, 7]. In ref. [6] the first working algorithm of a numerical simulation of the dynamically triangulated r a n d o m surface (DTRS) model was suggested, and in ref. [7] the first investigation of the model was made by means of a strong-coupling expansion.

Already in refs. [1,2], there was noticed the possibility of exact solution of zero-dimensional case of the model and the critical exponent ( ~ s t r = - - 1 ) of string susceptibility was calculated. Some additional results for the D = 0 case were obtained in ref. [8]. In refs. [6, 8] the case D = - 2 of the D T R S - m o d e l was solved exactly. The exact results were obtained for the string susceptibility (7~tr = --1) and for the mean-square extent in the embedding space: (X2)N CC log n, where n is the

" a r e a " of the surface (the number of triangles) which corresponds to the correlation radius exponent v~t r = 0. Some additional results concerning the D = - 2 case are contained in refs. [9-11].

C o m p u t e r simulations for the DTSP-model were performed in a n u m b e r of papers [6, 8,12-20]. Usually, they concerned the measurements of Ustr and Ystr for various dimensions.

An interesting development of the DTRS-models was the formulation and exact solution of some models of two-dimensional lattice quantum gravity interacting with various matter fields. The first exactly solvable model of this type was found in refs. [21, 22]. It was the Ising model on dynamical planar lattice (DPL), suggested in ref. [23]. This model is equivalent to the Majorana fermions interacting with 2 D - q u a n t u m gravity. The Majorana fermionic representation of the model was given in ref. [23]. The whole system of critical exponents was calculated in ref. [24].

In ref. [11] it was pointed out that the rather general case of matter fields preserving the principal possibility of exact solution are the Q-state Potts spins.

Besides the Ising model (Q = 2), the Potts models on D P L include the interesting cases of tree-like polymers (Q ~ 0), solved in ref. [11], and of the percolation ( Q - ~ 1) solved in ref. [25]. The particular limiting cases of tree-like polymers reproduce the exact solutions for D = 0, - 2 DTRS.

Another interesting exactly solvable case is the D = 1 model of D T R S which is equivalent to the quantum mechanical oscillator in the planar limit considered in ref. [26]. In what follows we shall pay special attention to this case and find the mass-gap exponent ~str = 0 which corresponds to the double logarithmic behaviour of the mean-square extent ( x 2) as a function of the area n of a surface

_ (log (1)

Together with the exact result Ystr--0 this sheds a new light on the critical properties of bosonic strings.


V.A. Kazakoo, A.A. Migdal / Noncritical strings 173 Some general remarks on the phase structure of the DTRS-models and the investigation of the limiting cases D ~ +_ ~ can be found in refs. [8,14].

Recently, great progress in the investigation of noncritical strings by means of conformal field theory was made. Polyakov has developed his rather old ideas [27]

concerning the Liouville theory of strings introducing an original light-cone gauge for the internal metric [28]. This makes it possible to obtain some new Ward identities in the corresponding conformal field theory and some new relations on the critical exponents of the theory. The missing ingredient, allowing the calculation of rather general series of critical exponents including ~str, Ustr and critical exponents of phase transition of Potts spins interacting with gravity etc., was found in the beautiful paper by Knizhnik, Polyakov and Zamolodchikov [29] (KPZ).

The most remarkable thing following from the comparison of both ap- p r o a c h e s - of lattice and conformal theory a p p r o a c h e s - is the complete coinci- dence of the results for critical exponents. This fact proves, in some sense, the correctness of both approaches.

In what follows we try to compare the old results for lattice models, as well as the new one for D = 1 DTRS, with the results of KPZ and to discuss the modern status of this theory. At first we give some general formulae of KPZ [29] for critical exponents.

(a) For the fixed " a r e a " A measured by means of the internal metric g,b(~)

The microcanonical partition function of random surfaces

has the asymptotics



ZD(A ) -- A-3+Y~"exp(const. A ) , (4) A ~

~str = l [ n - - 1 - t ( D - 1 ) ( D - 2 5 ) ] .


F r o m eq. (5) it is seen that ~str is real only for D < 1 or D > 25. The properties ot the theory in the interval 1 < D < 25 are, so far, unclear.


1 7 4 V.A. Kazakoo, A.A. Migdal / Noncritical strings

(b) The mean-square extent for D-dimensional random surfaces defined as 1

(x2)A -

A2Zo(A ) f

D / ( ~ ) f D g ( ~ ) 3 ( f d 2 ~ - A)

X f d 2 ~ ~ f dZ~'~/det

g ( l i ' ) [ x ( ( ) -

x(~')] 2,

is expected to behave as


(X2)A A -- a I log

A + a2 Avs,r


---~ OO

(c) For the Potts models on DPL, KPZ obtained the following formulae for corresponding critical exponents of spin-ordering phase transitions


~, for any Q,

1 - 2arcsin( )

[ ]_1

Ystr = 1 - a r c c o ~ ½ v ~ )



and for Ystr

2. Dynamical triangulation as a discrete approximation of 2-dimensional curved manifold

The basic idea of dynamical triangulation can be formulated as follows. Suppose we have some curved 2-dimensional manifold of a given topology described by a metric


~2) where ~1, ~2 are some coordinates on it. The internal metric is nothing but the "device" giving one ,the possibility to measure the minimal distance d S between two close points on the manifold

(dS)2 = gab d~ a d~ ~- (11)

To obtain the discrete approximation of this manifold with a metric

gab(~l, ~2)

one can imagine a surface made from adjacent equilateral triangles embedded into some auxiliary euclidean space of sufficiently high dimension. For a given surface (triangulation), we can say now that our discretized manifold consists of some


V.A. Kazakov, A.A. Migdal / Noncritical strings 175

number of discrete points (the vertices of a triangulation) and definite couples of these points (the neighbours connected by edges of a triangulation) are separated by the unit distance.

By increasing the number of triangles and choosing an appropriate type of triangulation we obviously can approximate a continuous manifold with any given accuracy. The coordinates 41, f2 and the metric t e n s o r gab(~l, ~2) can be regarded, respectively, as the local plane coordinates and the induced (by the above-men- tioned embedding) metric tensor. This calculation can be easily done by means of the technique described in ref. [4].

Any triangulation G can be unambiguously described by the adjacency matrix Gij, where i, j = 1, 2, 3 . . . . label the vertices

1, if i and j are theneighbours, (12) Gij = 0 o t h e r w i s e .

Of course, we consider only triangulations where any edge connects different points and any couple of edges cannot connect the same couple of vertices.

In the quantum theory of 2-dimensional gravity we should specify the summation weights for Gij in order to obtain the functional integration over the metric tensors g~b(~) in the continuum limit. This is the most delicate point in the formulation of dynamical triangulation: which should be these weights in the definition of the partition function? The natural answer to this question would be: just take all inequivalent triangulations with equal weights, like the Feynman graphs. But would this prescription lead to the same results as the sophisticated measures in the functional integrals of continuum theory, a n d / o r the algebraic quantization of conformal field theory?

If all these theories, including the dynamical triangulation, do make sense, they would be equivalent. But do they? That was a real question to which we may now answer, YES!

The first fact confirming this optimism is the result of exact solution of pure lattice 2D-gravity, which is the subject of sect. 3

3. Pure 2D-lattice gravity

The key to all exact solutions of the models of 2D-lattice gravity is the equiva- lence of triangulations to the planar Feynman graphs of @ theory. Just because each triangle has exactly three neighbours at any triangulation, the dual graph, which is obtained by connecting the middles of neighbouring triangles, is nothing but the planar qo 3 graph (see fig. 1).

The partition function of pure gravity reduces to the sum of all qo 3 planar graphs with the weight gn, where n is the number of q03 vertices (triangles on a dual lattice). This sum coincides with the vacuum energy of the q03 model

Zgrav(g ) = E vat(g2). (13)


1 7 6 V.A. Kazakov, A.A. Migdal / Noncritical strings

F i g . 1. A f r a g m e n t o f a t r i a n g u l a t i o n a n d its d u a l g r a p h .

The q03 coupling constant g is related to the cosmological constant of the gravity model. The expansion coefficients E~ ac of EVaC(g 2) have the meaning of the microcanonical ensemble partition functions. They can be read off from the classical paper [26]

2 (2n - 7)!

E,, = - (14)

n ( ~ n - 4 ) ! ( l n - 1 ) ! ' where n = 4, 6, 8,10 . . .

At large n this yields


E . . ~ M nV~" 3go ", (15)

_ _ 1 g2__ 3 (16), (17)

Y s t r - - - - 2 ' c - - ~ 6 "

The critical value of the coupling constant g¢ is not universal. It depends, say, upon the presence of tadpoles, self-energy graphs, etc. The above value corresponds to the eliminated tadpoles and self-energies.

The string susceptibility index 7str is universal. One may check that it remains the same for any cp'-theory in place of ¢p3.

Another good proof of universality is the investigation of the dependence of critical exponents on some parameters which are irrelevant in the continuum limit.

The most popular one is the a-parameter which corresponds to the adding of the following pure gravitational term S 1 to the lattice action S O


S g r a v = S 0 -t- S 1 ,

S O = (ln g ) n ,

S 1 =


qi = E Gij. (18)

i j


V.A. Kazakov, A.A. Migdal / Noncritical strings 177

T h e n u m b e r of neighbours q, is connected with the intrinsic curvature in the i th vertex

R~ = ~r(6/q i - 1), (19)

which tends to the Riemann curvature made from


In virtue of the G a u s s - B o n n e t theorem for a surface with a genus x


2 ~ r Y 2 R i ( ½ q i ) = 2 - 2 r . (20)


H e r e 1 3qi plays the role of an area element. The local limit of S O starts from the area t e r m

S O-~cOnst.fd2~ d ~ g , (21)

a n d S 1 f r o m the R z term

S 1 =

const, af

d 2 5 ~ R 2 ( 5 ) . (22)

This term is expected to be irrelevant in the sense of critical phenomena, i.e. the critical indices are expected to be a-independent, but a variation of a m a y cause some phase transitions.

As was proven in ref. [16] by the precise Monte Carlo calculations 7sDr =° is really i n d e p e n d e n t on a in the large interval - 2 < a < 10 (but for a < a c = - 2 the model a p p e a r s in the unphysical phase of " c r u m p l e d " surfaces with non-universal 3'str)- Accepting that, it was shown analytically in ref. [16] that

dYstr a~O

d a = 0 , for D = 0 . (23)

This meets our expectations coming from the field theory.

W h e n the prediction for 7str was obtained the field-theoretic approach could only give the W K B estimate from the 1 / - D expansion of the Liouville theory for D-dimensional strings [30]

Ystr(D) = ~ ( D - 7) + O ( 1 / ( - D ) ) . (24) The model of this section should correspond, in principle, to the D = 0 case. In fact, there was no systematic way to obtain the next terms of the 1 / D expansion until the exact formula (5) was recently discovered. At that time, it looked like eq. (24) was exact up to some nonperturbative corrections, so the discrepancy between the


178 V.A. Kazakov, A.A. Migdal / Noncritical strings

continuous field theory and the dynamical triangulations was puzzling. This was one of the rare events when the discrete model designed for numerical computations helped to cure the continuum theory.

Another interesting exactly calculable quantity for pure gravity (D = 0) is the probability distribution of the internal curvature (see eqs. (18) and (19)) for the infinite area n [8]

3 ) q (q - 2)(2q - 2)!


: 16 i 6

q ! ( q -

1)! q ~ e - q l n ( 3 / 4 ) . (25)

The exponential fall off implies that there are no critical fluctuations of internal curvature. Therefore a is not the relevant parameter.

4. Matter fields in lattice gravity

A natural way to introduce matter fields interacting with lattice 2D-gravity is to define these fields as the corresponding spins o i, i = 1, 2, 3 . . . . , in the vertices of a triangulation. The symmetry and integration (summation) measure of spins is to be chosen according to the physical problem and a local interaction between them should be chosen as a nearest neighbour interaction

5a(oi, oj)

on a graph G.

According to all our definitions the partition function of the whole model is

Z ( g , • , H ) = y'gn y"


- 2

±/3}2"~ ~<n)c~°¢

",kj ~ , ° k , oj) + H



] , (26)

n {G (n)} {G} [ k , j k = l

where the second sum is running over all triangulations with n triangles and the third one over all spin configurations. H is the magnetic field.

N o w we turn to the particular examples of matter fields - to the Q-state Potts models and the D-dimensional bosonic Polyakov strings.

5. Q-state Potts models on DPL: Some exact results for the Ising spins (Q -- 2), percolation (Q ~ 1) and tree-like polymers (Q --> 0)

In the Q-state Potts models on DPL defined in ref. [11] we choose the following definitions of ~ and f in eq. (26)

5a(ok, oj) = 6okoj - 1, f(ok)

= ~ l , a k - 1, (27), (28) where o k = 1,2 . . . Q. As was shown in ref. [11],


fl, H = 0) defined by eqs.

(26)-(28) can be expressed in terms of planar (large-N) limit for the following


V.A. Kazakov, A.A. Migdal / Noncritical strings 179 matrix integration model

F Q ( g , ~ , H = O ) = Nll m - - 1 0 ~ - - - 1 I d N X e - N t r X - 2

N 2 $ J

X ( f d N 2 U e x p N t r [ h X U - ½ U 2 + l ~ , U 3 ] } O, (29)

where X and U are N x N hermitian matrices, and

h 2 = [ e B + Q _ l ] 1, ff2 = if(/3, Q ) g 2 . (30), (31) It can be proven by performing the gaussian integral over X (after the introduction of integrals over Q different matrices U 1 ... UQ) and by expanding in g. According to the usual Feynman rules, every coefficient F(n)(/3) will be equal to the corre- sponding coefficient in eq. (26) after the duality transformation, changing the q~3 graphs by dual triangulations, spins by dual spins in the vertices of triangulations, and the inverse temperature/3 by the dual one

The angular representation

X = ~0+x~o, U = f2+M2, (33)

where ~o and 9 are SU(N)-group matrices and x=diag(xa,...,XN) , ~=

diag(kl . . . XN), and the integration in eq. (29) over the q0 = (~2~o +) group variable by means of the formula [31]

N - I

) det [ehX,~/]

f(dep)SU(N)exp[htr(q)+)~q)x)] = H p! h - N ( N - 1 ) / 2 0'

a ( x ) a ( x ) ' (34) where A(z) = Hi> j(zi - z j) is the Van der Monde determinant, leads to a represen- tation in terms of the eigenvalues )~i and x i [11]

1 N



l i r a ~ 5 1 o g

× = dX exp[-N( X + j (35)


180 V.A. Kazakoo, A.A. Migdal / Noncritical strings TABLE 1

Critical exponents for the Ising model on a dynamical planar lattice (DPL) and on a regular 2D-lattice

Ising model

Critical Ising model on a regular lattice

exponents on DPL [22-24] (Onsager solution)

a - 1 0

/3 1_ 2 ! 8

8 5 15

vd 3 2

"/str _ 13

In principle the problem is reduced to some saddle-point calculation, but this calculation does not seem to be trivial. The complete solution would be very interesting from the point of view of comparison with the results (8)-(10) of the K P Z theory. It would be the first example of exactly solvable string theory with the matter fields having a continuously varying central charge (see ref. [32]) (not only in a phase transition point). But this solution is so far absent.

We now consider only three particular cases which are explicitly solved.


In this case the problem is reduced to the so-called two-matrix integral [21, 22]

which was calculated in ref. [33]. In ref. [24] the model was investigated in presence of a magnetic field and the properties of the spin-ordering phase transition were established. It appeared that the phase transition is third order ( F ' " (/3) is discontin- uous) and the critical exponents differ from those of the Onsager solution (see table 1). All the exponents satisfy the usual scaling relations and are calculated indepen- dently (unlike the magnetic field exponents in the usual Ising model on a regular lattice). The Ystr exponent is equal to -- ~ only at the critical point /3 = tic. For any o t h e r / 3 , Ystr = -- ½ as in pure gravity. It shows that the corresponding Majorana fermions on D P L describing the Ising spin excitations are in the long-wave regime for/3 =/3c-

All the results of table 1 completely agree with those of the K P Z conformal field theory for Majorana fermions interacting with 2D gravity [29] (see formulae (8)-(10) for Q = 2).

5.2. Q ---, 1: P E R C O L A T I O N O N A R A N D O M P L A N A R L A T T I C E

It is shown in ref. [11] that in this limit the Potts model on D P L reduces to the classical bond-percolation problem with the probability of the bond to be conduc- tive being

p = l - e - # .


KA. Kazakov, A.A. Migdal / Noncritical strings 181

But the planar lattice should be chosen at random with the statistics of all planar graphs entering the ensemble with equal weights (as in the pure lattice gravity of sect. 3). In the thermodynamical limit we can use only one randomly chosen graph G (~) considered as a lattice with quenched defects. The corresponding partition function


p )

FPerC(p) = lim lim (36)

n ~ Q ~ I


1) '

appears to be equal to the mean number of percolative clusters per unit volume [11].

This quantity has been recently calculated in ref. [25]. The result shows that near the percolation transition point Pc


Fperc(P) -


P)2-'~ l ° g ( P c - P ) ,


a = - 2 , (38)

in remarkable agreement with the formula (9) of the KPZ theory.

5.3. Q --* 0: T R E E - L I K E P O L Y M E R S O N D P L

According to ref. [11] by the choice

h 2 = (39)

Q(B +

1) '

in eq. (29) we obtain, in the limit Q ~ 0, the partition function of tree-like polymers on D P L

ztree(g, B) = E f t n E B y~tree), (40)

n { G ~ ) } trees o n G~m

where the internal sum goes over all tree-like configurations (collections of clusters without loops) on G ~"), and oW(tree) is the size of a given tree-like configuration (number of edges in it); g is proportional to g (see ref. [11]) and In B is the chemical potential of trees. In the limit B ~ 0 there are no trees on G ~n) and we come to the problem of pure gravity considered in sect. 4. In the limit B ---, oo only maximal (spanning) trees survive and we get the model with

~ s t r = - 1 , (41)

This model is nothing but the D = - 2 case of the Polyakov string considered in sect. 6.

So, there exists the phase in this model for sufficiently large B, which is characterized by eq. (41). Again this result coincides with the Q---, 0 limit of the


182 V.A. Kazakov, A,A. Migdal / Noncritical strings K P Z - t h e o r y formula (10).

6. Discretized Polyakov string

In the case of a bosonic Polyakov string in the D-dimensional embedding space we choose the spins a i as arbitrary D-dimensional vectors

o, =- x~, /.t = 1,2 . . . . D. (42)

The whole action Sstri~g is chosen as

Sstfing = Z G i j ( x i - x j ) 2 q- (const.)n (43) i , j

T o this action we can add the irrelevant term S 1 (see eq. (18)) as well The first term in eq. (43) is the discrete analogue of the gradient term and the second one represents the cosmological term. This action in the continuum limit tends to the usual Polyakov string action

s-, f d2~(g°haox~abx~+const,).


As we have noticed in sect. 2 the continuum limit (44) can be derived by means ot technique of refs. [4, 5] by limiting the consideration only by equilateral triangles.

There is no real problem with the integration measure for the xi fields since any auxiliary local factor in the measure Dx can be absorbed into the triangulation- dependent part of the action (i.e. into the pure gravity part). So we simply write

Dx = 1-I dDxi •



N o w the definition of the discretized Polyakov string (DTRS-model) is complete. In the rest of this section we review the numerical and analytical results of this model and compare them with the KPZ theory. Some new exact results for the D = 1 case of the model are given in sect. 7.

The majority of the Monte Carlo experiments and strong-coupling calculations for the DTRS-model are devoted to the measurement of two critical exponents, "/str and Gtr (DH =


is the Haussdorff dimension of embedding).

At first Dla was measured for D = 3 in ref. [6] by means of the original Monte Carlo procedure which used the simultaneous updating of x i and G ~") degrees of freedom of the model, and then for some other values of D in a number of papers [8,12-20]. The measurements concerned the mean square size (x2)n of the random surfaces with the fixed " a r e a " n of triangulations

11( )

(x2)~ n ( n 2 - 1 ) D

E ( X i - - X j ) 2 .


l, J 71


V.A. Kazakov, A.A. Migdal / Noncritical strings 183

All numerical results lead to the finite Haussdorff dimension in the interval D - 0 - 8 , which means that

( x 2 ) , , _ .2.s,r, Dr~ = 1/p~t r--- 8-10, for D - 0 - 8 . (47)

F o r D --* 0 the analytical continuation was used, leading to a combinatorial formula for eq. (46) (see ref. [8]).

The statistical errors are small enough, and the fit of the power law is quite convincing. The logarithmic law fits numerical data much worse. The results does not agree with the predictions (7) of the K P Z theory, which yields the logarithmic law at any D < 1 (for D = 1, see at the last c o m m e n t of sect. 7). There m a y be several explanations of this discrepancy.

(a) The systematic errors coming from finite-n effects.

(b) The spurious fixed point of the renormalization group influencing the discrete model.

(c) There is the operator with a dimension 2pst r ~ 0.2 in the corresponding c o n f o r m a l theory which has to be included in the x x operator product expansion.

We cannot completely rule out the first explanation inspite of the fact that in some experiments the number of triangles n exceeded one thousand. The possibility of systematic errors was discussed in refs. [18,19] on the basis of a high-statistics M o n t e Carlo measurement as well as of the data of strong-coupling expansion p r o p o s e d in refs. [7]. So we have to leave open this important problem.

T h e calculation of Tstr represents a more subtle problem for numerical methods because it is concerned with the determination not only of the leading exponential term (4) but also of the subleading one. The first results for Tstr by various D were obtained in ref. [7] by means of strong-coupling expansion (except of exact results for D = 0 and D = - 2). Agreement with the K P Z theory appears not to be bad for D ~< 0. F o r D = 1 the discrepancy is larger. We think that the problem is the inverse logarithm correction predicted in sect. 7.

As for the Monte Carlo calculation of 7str there exist two different algorithms.

One of them is the canonical simulation which uses the varying number of triangles n. In refs. [17-19] the high-statistics measurements were performed by this method.

It is interesting that the results of refs. [17] qualitatively agree with the K P Z predictions for D < 1 and D > 25, but the authors complain of the large systematic errors due to the finite-size effects.

A n o t h e r method uses the microcanonical ensemble to calculate the ratios r n =

Z n a / Z , as special averages over this ensemble. It was suggested in ref. [16] and

successfully used there for D = 0 calculations of 7st~- In ref. [20] this method is used for some other dimensions, not only for the model in question but also for summing up planar graphs in the usual scalar qo3-theory. These two theories appear to have rather similar behavior as regards 7~tr(D). Preliminary results of ref. [20] for n < 44 show that in the interval 1 < D < 4 Tst~ seems to be zero within the statistical errors.


1 8 4 V.A. Kazakov, A.A. Migdal / Noncritical strings

If proved for larger values of n, this fact would be an interesting prediction for the interval 1 < D < 25 still missing in the K P Z theory.

Let us turn to some analytical results for DTRS. The first is already given by eq.

(16) for D = 0. N o w we consider another exactly solvable case, D = - 2 . The analytic continuation to negative D is obvious, since the integral over x-variables is gaussian

" [ ]

f ,I~= 1 dnxk6(xl)exp - E Gij(xi- xj) 2

= [ d e t ' ( L i j ) ] - D / 2 ,

- i , j


where n v = i n + 2 is the number of vertices, Lij = 8ijqj - Gij, and det' stands for the (n v - 1) × (n v - 1)-diagonal minor of Lij (all diagonal minors coincide).

At D = - 2 the determinant appears in the first power which allows one to use the combinatorial methods to compute the sum over graphs. The basic relation is the Kirchgoff formula

d e t ' ( L ( G ) ) = E 1, (49)

m a x . t r e e s o n G

where max. trees on G means the spanning trees made from the qo 3 graph G by cutting all the loops. The detailed definition and solution of the model can be found in the original papers [6, 8].

Later the model was reconsidered by large-N methods [10]. As was noticed in sect. 5 this model can be viewed as a limiting case of Potts models on D P L which gives another alternative possibility to solve it [11].

T h e solution gives the predictions for "/str as well as for Pstr

[6, 8]

Ystr = - - 1, (for spherical topology),


Vst r = 0, ( ( x 2) - l o g n ) . (51) Both predictions agree with the formulae (5) and (7) of the K P Z theory. The f o r m u l a for '),st r was generalized for the surfaces of the arbitrary genus g [34]

7 n=°str -- ~ ( g - 1), (proven for g = 0 , 1 , 2 ) , (52) ystnr= -2 = 2 + 3 ( g - 1), (for all g ) .


T h e linear growth of 7sir with genus implies the divergence of the sum over topologies for bosonic strings.

T h e c o m p u t e r results at D = - 2 [17-19] agree with eq. (50) but with larger errors t h a n at D = 0. As for the logarithmic law (51) there are slight discrepancies but no


V.A. Kazakoo, A.A. Migdal / Noncritical strings 185 real contradictions unlike the case D = 0. The statistics are worse at D = - 2 , since the determinants (48) should be explicitly calculated for every updating of graphs.

In ref. [6] it is argued that for the limit D ~ - oo the regular graph G O survives in the sum over graphs and for this graph the same result (51) is true as for the D = - 2 case. So we get again the coincidence with the K P Z theory.

7. Discretized Polyakov string for D -- 1: S o m e exact results

Let us n o w turn to the most interesting case of the lower critical dimension D = 1. The planar graphs for D = 1 were summed up already in the pioneer p a p e r [26], but until recently we did not take this solution seriously since it corresponded to the F e y n m a n propagators [ ( p , - p j ) 2 + 1]-1 instead of the gaussian propagators of the D T R S model.

However, there are no ultraviolet divergences at D = 1 (this is just the q u a n t u m mechanics of the N × N matrix coordinate in the limit N ~ ~ ) . So the universality p r o p e r t y of the critical phenomena allows us to expect the same critical exponents for the F e y n m a n propagators as for the gaussian ones. Some numerical verifications of this conjecture can be found in ref. [20].

T h e solution of the D = 1 model is so simple that we m a y reproduce it here for the sake of completeness.

T h e hamiltonian for the hermitian matrix model reads


H = N - ½ . . 0~o-~Tj + trU(~o)

U ( f~ ) = 1 2 1 _fp3 7cP + 7g ,

but it could be more general. The corresponding variational principle

J ,

3qJ ]2+~2NtrU(~p)],





f!-! d%j 2 (58)


N o w consider the invariant ansatz, depending on the eigenvalues 7~ 1 .. . . . X N of the hermitian matrix ~ij

~ = l p ( X l , ~k 2 . . . A N ) , ( 5 9 )



V.A. Kazakov, A.A. Migdal / Noncritical strings

and eliminate the angular variables $2 in q~ = ~2+ML This yields the well-known Van der Monde determinant


A(X) = I-I ( x , - h i ) ; (60)



= d(~2)SU(N)A2(X)H dXk- (61) k

N o w we absorb A(X) in the test wave function

a(x)¢(x) =x(X),

( ~ ) = f l ~ I k d X k × a ( x ) , (62),(63)

and transform the kinetic term as follows

Vsu(N) f H dXkd(n)su(u) Y.

1 0tP ]2A2(~,)

i, j O•ij J

Z o~'~ ] 2A2( ~ )

=fHdXk (~,^,!


O l n A ( h ) ) 2

= f H d x , 5X~, -x ax,


Y£,j - 2 x - -


OX 0 l n A ( X ) 0X i OX i

'" ']

3Xi + 3Xi , (64)

where VSU(N ) is the volume of the group space. The second sum vanishes identically so that we are left with the trivial invariant hamiltonian

__Hi,,,= 1 + U(X,) . (65)

N 20X~


V.A. Kazakov, A.A. Migdal / Noncritical strings 187 This is just the collection of the independent one-particle terms, but with the Fermi statistics instead of original Bose statistics! This observation was one of the pearls ot the famous papers [26]. The wave function X()t) is antisymmetric due to the factor


The rest of the construction is just the standard analysis of the ideal Fermi gas. In the relevant W K B approximation (which holds at N ~ ~ ) , we find

EVAC= fdXdp

( p 2

0(OF p2


dXdP o( e p2 )

N = f 2~r k F 2 N U()k) , (67)



is the step function. The density of levels p = d e F / d N , which defines the gap in the (radial part of) spectrum, reads

ON [d2tdp{ p2 )

l / p 0£F g ~ l £ F - 2---N - U(2t) . (68)

N o w it is convenient to rescale )t ~

h/g, U()t )

---, U ( h ) / g 2, ('F =

Nl't/g 2, P



which eliminates N, g from the above integrals

EVAC 1 dXd~

- g4 f ~ . _ _ (1~2 .~_ U(X))0(~- 172- g(~k)),


~ (69)

" d X d ~ 0 ( ~ 2

g Z = j ~ t ~ - - - U ( X ) ) ,


2 r d ) t d ~ . 172

1/p=g J ~ 3 ( t ~ - ~ -U(X)).


These integrals possess singularities when ~t coincides with any local maximum of


U'(~.o) = 0, #c = U ( k o ) , U"(Xo) < 0. (72) The nature of singularity can be found using the obvious relation

0 ( EvAc)_ #

3 g 2



188 i.e.

V.A. Kazakov, A.A. Migdal / Noncritical strings

0 [ E V A C

\ | g 4 ) = ~.~.

Og 2 (74)

The singularity of the function /~(g2) follows from eq. (71), (73)

1 1

(g2p'-""~ ~--'t~:

2 7 r ¢ _ U , , ( X o ) I n ~-C_~- c , (75)

g2 _ g~ _ (/~c - / t ) l n ( / ~ - / t c ) , and finally, from eq. (74)


(3g2) 2

~E vAC - ~g2~/t ln(Sg)2 . (77)

We find the surprising result: the specific heat diverges as

O 2E VAC 1 1

C . (78)

O(g2) 2 In ( 3 g 2 ) l n ( n )

This agrees with the general prediction 7str = 0, but the 1 / l n ( n ) law has yet to be checked within the KPZ theory.

Another puzzling prediction of our model is

( x 2) > 1/02 ~ ln2(g2 _ g2)


We do not know the angular excitations, but those could only lower the gap, so we arrive at the above inequality. This is compatible with the KPZ theory, as is clear from the formula

nO- l - o l n n

, In 2 n. (80)

2 o2 o s o

It is interesting that for D = 1, the formulae of ref. [29] for operator dimensions Am, n give the equidistant spectrum as is the case for the eigenvalues part of the spectrum of the D -- 1 DTRS model of sect. 5. But the role of angular excitations is so far unclear. The numerical results of ref. [20] show that not only the asymptotics


E v a c l o g ( n ) ' (81)


V.A. Kazakov, A.A. Migdal / Noncritical strings 189

which follow from eq. (54), fit well the experimental data, but the asymptotics

(x25, - l o g 2 ( n ) , (82)

(and not { x 2 ) , - log(n)) following from eq. (79), satisfy the experiment in the best way. So it seems that the angular excitations do not influence the lowest excitation (as is true for the hydrogen atom) and the inequality (78) should be replaced by the asymptotical equality (82).

T o summarize, some interesting phenomena occur at D = I in the string theory.

The field-theoretical interpretation of logarithmic laws is yet to be clarified but the critical exponents ~'str and ust r agree with the KPZ theory.

8. Conclusion

As we have seen from this paper there are practically no disagreements between the continuum KPZ theory and the lattice version of quantum gravity based on dynamical triangulations. Of course, the KPZ approach can now be used to deal with more general types of matter fields than lattice approach, but the following features of the latter are attractive

(a) For some models (such as Q = 0,1, 2 Potts models in the magnetic field) the lattice approach makes it possible to solve them not only at the phase-transition point and zero field unlike in the KPZ approach.

(b) The lattice method is perfectly suited to numerical methods which are necessary in such unclear cases as the 1 < D < 25 interval in the bosonic string.

(c) The large-N methods can in future give an approach to 2D-gravity as general as the Baxter transfer-matrix approach to the two-dimensional field theories on a regular lattice. Such an approach would considerably supplement the conformal theory methods.

It is apparent that many questions remain to be answered than those presently solved by both approaches.

We are grateful to A.M. Polyakov, A.B. Zamolodchikov and D.V. Boulatov for m a n y valuable discussions. One of us (V.K.) wishes to thank the Niels Bohr Institute for kind hospitality.

Note added in proof

Recently, the K P Z results were reproduced in the conformal gauge, i.e. within standard Liouville theory (Distler and Kawai [35] and David [36]). The genus dependence of 7, found in these papers, agrees with our formulae. Our results for


190 V.A. Kazakov, A.A. Migdal / Noncritical strings

D = 1 are also confirmed. The more precise measurements of v, using supercomput.

ers a n d / o r dedicated machines, are necessary anyhow.

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